I'm currently running to lose weight and build some fitness. I run 12 km run in about 1h20m and keep my heart rate between 140~150bpm. I try to do 3 runs a week. The run itself does not seem very stressful but should I be eating extra protein to repair damage. Diet-wise, I am a pesco-veg who eats a few eggs and a couple of portions of fish a week.

Extra details requested: I'm 43 yo, 184cm, 90kgs and about 18% BF. I've have been running for a few months now. Years ago, my heart rate used to hit up to 180bpm while running so I would consider 150bpm to be a lot more healthy. My polar watch has this as my best weight loss zone though I'm trying to up it a bit just so I can get the run done faster. I have a lot of dairy as it is (2 or 3 300ml lattes a day, cheese and yoghurt).

The main question is: does a light(ish) exercise regime like this warrant extra protein. When I was weight lifting 20 years ago, I'd ensure I had extra protein to ensure maximum bulk. Now, I just want to lose the gut.

yes certain type of protein according to dairy industry like here (marketing whey for all kind of things such as weight loss, surely-better-than-soft-drinks -style argumenting, feels a bit strawman) but if you get enough things already from your nutrition, probably not. Be on the safe side though some extra protein such as whey straight after running is surely not bad, some fuzzy question here.

0. Calculations and Profile analysis

With 90kg, it is apparently around 135g:

1.5*90
[1] 135

but but -- is this over-estimate in your situation? Researching.

If you think that two fitness-spoons is about 30g so some protein probably not bad, depending ofc on your diet and intensity of your exercise (HF does not tell everything, EPOC more, there are services such as Movecount to monitor this kind of things). Now other type of proteins such as egg and soy. Egg weights about 50g and has about 12% protein, source. In soy about 52% of content. In whey about 76% or even more of content. And also take into account your other intake during day, can you get protein already from your nutrition?

1. Focus on general picture not on one type of thing such as protein or vitamin

Only one type of thing such as protein or calcium is not necessary. It is necessary that you get the all necessary ingredients to get long chain reactions in your body working. The reason why the next study focuses on both Ca and D2 is that absorption requires them, not just one. The quote from this study about the elderly women here:

There are also other things such as C that may affect things -- so at that age just take some easy to get multivitamin without confusing your head too much with technical details or rationing, even the term "vitamin" is a pain of history!

2. Eat good quality of protein with variety

The other answer with the generalization "Protein always helps with recovery." is wrong and misleading. The sentence is satisfied even with gelatin, not including all necessary amino acids. Again the key is "variety". If somethnig feels bad, stop. Mix between egg, soy, whey and milk.

The evidence for post-workout protein shakes increasing your performance for running isn't conclusive, so this is somewhat anecdotal advice. I've been running for about 4 years regularly and in the past 6 months tried protein shakes after the long and mid runs. I've increased my speed from 8.1mph to 9mph in that time, and up to 9.5-10mph for 5k runs.

I've also lost weight in that period from the eating less via the protein shakes, A 500ml serving keeps you sated for an hour or so later until a main meal, and is about 120 calories.

The protein powder I used is by Bio-synergy, which is whey powder (basically dried milk). The cheaper stuff you get at a discount seems to be missing not only taste, but some amino acids and bulked out with soya. The one chose is also targeted at lean muscle growth rather than body building/bulking up, which helps.

It's an easy and fairly cheap experiment to try, just be sure to stick with a good brand that is carb free, and avoid the soya shakes.